Optica Applicata, Vol. XLII, No. 1, 2012
DOI: 10.5277/oa120109
The transmission characteristics
under the influence of the fifth-order
nonlinearity management
XIUJUN HE1, 2, KANG XIE1
1
School of Optoelectronic Information, University of Electronics Science and Technology of China,
Chengdu, 610054, China
2
Chengdu University of Information Technology of China, Chengdu 610225, China
Starting with the nonlinear Schrödinger (NLS) equation, we have derived the evolution equations
for the parameters of soliton pulse with propagation distance in optical fibers, taking into
consideration the combined effect of second-order dispersion and the fifth-order nonlinearity by
means of variation method. According to nonlinear evolution equations, the evolution of the pulse
width with propagation distance is obtained under the influence of the different fifth-order
nonlinearity. The results show that the pulse width fluctuates periodically under the influence
of the different fifth-order nonlinearity. In the cycle, the negative fifth-order nonlinearity makes
the pulse-width greater than the initial value while the positive fifth-order nonlinearity makes
the pulse width less than the initial value. However, under the positive and negative fifth-order
nonlinearity management, compared to the impact of positive or negative fifth-order nonlinearity
only, the fluctuations of the solitons width are greatly reduced, even disappear. In other words,
the width maintains almost steady. Therefore, it is possible that the pulse width is to be transmitted
without any deformation.
Keywords: fifth-order nonlinearity management, soliton, nonlinear Schrödinger equation, variation
method.
1. Introduction
The optical pulse compression and soliton transmission mode have been widely
cosidered by researchers [1–3]. At present, under a third-order nonlinearity, the transmission of optical pulse has been intensively investigated, but with the development of
high-nonlinear fiber (such as semiconductor doped fiber, organic polymer fiber, etc.),
fifth-order or higher order nonlinearity can occur in the medium power of light [4].
Therefore, researches began to study the self-phase-modulation, the modulation
instability [5], the solitary wave transmission, the bistable behavior [6], ground-state
soliton, the propagation properties of the Gaussian pulse in the case of the fifth-order
nonlinearity [7]. However, it is still a new field for research on transmission of higher
104
XIUJUN HE, KANG XIE
order soliton. With the use of variation method and from the extended nonlinear
Schrödinger (NLS) equation of the fifth-order nonlinearity, this paper introduces
evolution equations of soliton parameters depending on the propagation distance under
second-order dispersions and cubic-quintic nonlinearity, simulates the evolution of
soliton width on the propagation distance. Nowadays, researchers make greater efforts
to apply nonlinearity management to control the transmission of soliton, as with the
dispersion-managed soliton propagation [8, 9]. The authors of the present paper focus
on effects of the fifth-order nonlinearity management on the width of soliton and their
achievements so far appear to be very helpful when investigating the transmission of
soliton.
2. Theoretical analysis
Let us start with the nonlinear Schrödinger (NLS) equation with the fifth-order
nonlinearity [5]:
β 2 ∂ A ( z, t )
∂A ( z, t )
- + γ A ( z, t ) 2 A ( z, t ) + γ ' A ( z, t ) 4 A ( z, t ) = 0
i ------------------------- – ----------- --------------------------2
2
∂z
∂T
(1)
2
where i = (–1)1/2 and A(z, T ) is the slowly varying envelope of the optical pulse and
T the temporal coordinate frame that moves at the group velocity νg of the pulse,
and z the spatial coordinate representing the transmission distance, β 2 represents
the second-order dispersion parameter, γ and γ ' are respectively the nonlinear Kerr
effect coefficient and the fifth-order nonlinearity coefficient.
We proceed to the normalization of Eq. (1) in the following way:
2
s ∂ u
∂u
2
4
i ------------- + -------- --------------+ g1 u u + g2 u u = 0
2 ∂τ 2
∂ξ
(2)
Here, u = A / P 10 ⁄ 2, ξ = z/LD, s = –sgn( β2), τ = T /T0, g1 = LD / L NL1, and g2 = LD / L NL2,
while LD = T 02/|β2 |, L NL1 = 1/(γ P0) and L NL2 = 1/(γ ' P0).
Variational method was used to solve Eq. (2) and the Lagrangian density is
obtained from Eq. (2),
i
s
L = – -------- ⎛uu *ξ – u * u ξ ⎞ – ------- u τ
⎝
⎠
2
2
2
g1
+ ---------- u
2
4
g2
+ ----------- u
3
6
(3)
The following generic ansatz for the stationary solution of Eq. (2) shall be
considered
2
u = asech r ( t – q ) exp i ϕ + ic ( t – q ) – i ω ( t – q )
(4)
The transmission characteristics...
105
where a, r, q, ϕ, c and ω represent the pulse amplitude, width inversion, central
position, the phase, chirp and frequency shift, which are all real functions of ξ
δ ∫ 〈 L 〉 dξ = 0
(5)
〈 L 〉 is the average Lagrange density function, that is,
〈 L〉 =
+∞
∫–∞ Ld τ
=
2
a ⎛ 1 2 dc
dq
1 2 2
1 4
2 2
2
= ---------⎜– -------- π ----------- – -------- π c – r ω – 2r ω ------------ – --------- r +
3
dξ
dξ
3
3
r ⎝ 6
2
16
2 2
2 4
2 dϕ ⎞
+ --------- g 1 r a + ------------ g 2 r a – 2r -------------⎟
3
45
dξ ⎠
(6)
From Equation (5), the parameter variation equation can be obtained
d 〈 L〉
------------------ = 0
dy i
(7)
where yi represents parameters a, r, q, ϕ, c and ω. Parameter evolution equations can
be derived by applying Eqs. (6) and (7)
da
-------------- = – asc
dξ
(8)
dr
-------------- = – 2rsc
dξ
(9)
dq
-------------- = – s ω
dξ
(10)
dω
-------------- = 0
dξ
(11)
1 ⎛ 4
32
2
2 2
2 4
dc
- 2r s – 2g 1 r a – ------------ g 2 r a ⎞
-------------- = – 2sc + ----------2 ⎝
⎠
15
dξ
π
(12)
1 2
5
32
1 2
2
4
dϕ
-------------- = -------- r s – -------- g 1 a – ------------ g 2 a – -------- ω s
3
6
45
2
dξ
(13)
106
XIUJUN HE, KANG XIE
3. Computational results and analyses
3.1. Soliton propagation in different fifth-order nonlinearity situations
Using the above evolution equations of soliton parameters, the change of soliton width
is as shown in Fig. 1. So, here g1 = 1, s = 1 and initial values of solitons, a0 = 1, r0 = 1,
c0 = 0. When g2 < 0, the pulse width changes in a periodic and fluctuating way and it
is greater than the fluctuation of initial width, that is to say, when the initial pulse width
starts to increase gradually, eventually increasing to the soliton width peak, it begins
to gradually diminish, eventually reducing to the initial pulse width and starts to
increase again. Then, these changes continue to repeat. The larger the | g2 | is, the longer
it lasts, and the more times it is repeated, the larger its fluctuation is. When g2 > 0,
the pulse width also changes in a periodic and fluctuating way and it is smaller than
the fluctuation of initial width, that is to say, when the initial pulse width starts to
decrease gradually, eventually decreasing to the lowest value of the width, it begins
to gradually increase, eventually increasing to the initial pulse width and starts to
decrease again. Then, these changes continue to repeat. The larger the | g2 | is,
the shorter it lasts, and the more times it is repeated, the larger its fluctuation is.
As for the equivalent positive and negative fifth-order nonlinearity, the negative fifth-order nonlinearity of influence on the pulse width is much bigger than the positive
fifth-order nonlinearity.
3.2. Soliton propagation under the fifth-order nonlinearity management
We know that the dispersion management lowers mean dispersion of the whole
transmission line through configuring properly the fibers with opposite dispersion
Pulse width [ps]
5
4
g2 = 0.2
g2 = 0.4
g2 = 0.6
g2 = –0.2
g2 = –0.4
g2 = –0.6
3
2
1
Pulse width [ps]
0
5
4
3
2
1
0
0
10
20
ξ
30
40
50 0
10
20
ξ
30
40
50 0
10
20
ξ
30
40
50
Fig. 1. The evolution of the pulse width with propagation distance under the influence of the different
fifth-order nonlinearity.
The transmission characteristics...
107
properties in order to improve transmission performance of solitons. As to the dispersion management principles, nonlinear coefficient is also a function of ξ, which shows
periodic change along the fiber length. In a cycle, one situation of nonlinear coefficient
is that absolute values are identical but opposite in sign, the other situation is that
absolute values ranges and the opposite is the sign, which is called nonlinearity
management. As for the fifth-order nonlinearity management g2(ξ ) should satisfy
the following equation, that is,
nL ≤ ξ < l 1 + nL
⎧ g+
g2 ( ξ ) = ⎨
⎩ g–
l 1 + nL ≤ ξ ≤ ( n + 1 )L
,
n = 1, 2, ...
(14)
Here, g+ > 0, g– > 0, and L = l1 + l2 are the cycle lengths of the nonlinear management;
l1 and l2 are the interaction lengths of the positive and negative nonlinearity. Obviously,
the positive nonlinearity makes the pulse width decrease and the negative nonlinearity
makes it increase. Therefore, using the positive and negative nonlinearity management,
their functions can cancel out, which improves drastically the fluctuation of width.
At first, when g+ = g– , l1 = l2, the soliton transmission in the fifth-order nonlinearity
management is shown in Figs. 2 and 3. Comparing to Fig. 1, the fluctuations of
the soliton width reduce greatly after applying the fifth-order nonlinearity management. The negative fifth-order nonlinearity develops the soliton width above and
Pulse width [ps]
5
g+ = g– = 0.2
4
g+ = g– = 0.4
g+ = g– = 0.6
3
2
1
0
0
10
20
ξ
30
40
50 0
10
20
ξ
30
40
50 0
10
20
ξ
30
40
50
Fig. 2. The evolution of the pulse width with propagation distance under the influence of the fifth-order
nonlinearity management (L =l1 + l2 = LD).
Pulse width [ps]
5
g+ = g– = 0.2
4
g+ = g– = 0.4
g+ = g– = 0.6
3
2
1
0
0
10
20
ξ
30
40
50 0
10
20
ξ
30
40
50 0
10
20
ξ
30
40
50
Fig. 3. The evolution of the pulse width with propagation distance under the influence of the fifth-order
nonlinearity management.
108
XIUJUN HE, KANG XIE
5
Pulse width [ps]
15
g+ = g– = 0.8
I1 = I2 = LD
10
g+ = g– = 0.8
I1 = LD
I2 = 0.7LD
4
3
2
5
1
0
0
10
20
ξ
30
40
50
30
40
50
0
0
10
20
ξ
30
40
50
Pulse width [ps]
5
g+ = 0.8
g– = 0.41
I1 = I2 = LD
4
3
2
1
0
0
10
20
ξ
Fig. 4. The evolution of the pulse width with propagation distance under the influence of the fifth-order
nonlinearity management.
beyond its initial value fluctuation, while the positive fifth-order nonlinearity develops
the soliton width to be less than its initial value fluctuation and they can be cancelled,
which decreases drastically the fluctuation. Better results are achieved by using
a smaller cycle.
In Figure 4, due to dissymmetry of the function of positive and negative fifth-order
nonlinearity, an effect of the fifth-order nonlinearity management is not good enough
when the fifth-order nonlinear coefficient is bigger under the condition of g+ = g– and
l1 = l2. According to the function of equal negative fifth-order nonlinearity above and
g2 = 0
g2 = –0.6
g2 = 0.6
50
Distance
40
30
20
10
0
–5
0
Time
5 –5
0
Time
5 –5
0
5
Time
Fig. 5. The evolution of the soliton with propagation distance under the influence of the different
fifth-order nonlinearity.
The transmission characteristics...
109
g+ = g– = 0.2
g+ = g– = 0.4
g+ = g– = 0.6
50
Distance
40
30
20
10
0
–5
0
Time
5 –5
0
Time
5 –5
0
5
Time
Fig. 6. The evolution of the soliton with propagation distance under the influence of the fifth-order
nonlinearity management.
beyond the positive fifth-order nonlinearity and under the condition of g+ > g– and
l1 > l2, choosing a suitable ratio we can get a good performance through the fifth-order
nonlinearity management.
Finally, the soliton transmission as a function of the fifth-order nonlinearity is
calculated by applying numerical methods. The results are consistent with those of
theoretical calculations, see in Figs. 5 and 6.
4. Conclusions
This paper applies the variation method to research effect of the fifth-order nonlinearity
on the soliton transmission. The result shows that pulse width of soliton has a cyclical
fluctuation under the influence of the fifth-order nonlinearity. The negative fifth-order
nonlinearity develops the soliton width above and beyond its initial value fluctuation
while the positive fifth-order nonlinearity develops the soliton width to be less than
its initial value fluctuation in a cycle, and the bigger the absolute value of the nonlinear
coefficient, the greater the fluctuation. On this basis, the influence of the fifth-order
nonlinearity management is further researched to find out that the fluctuation of
soliton width is greatly reduced under such conditions, and the soliton width is almost
not changed, which can make pulses realize transmission of original shape under
the function of the fifth-order nonlinearity.
References
[1] AGRAWAL G.P., Nonlinear Fiber Optics, Fourth Edition, Academic Press, US, 2007.
[2] HASEGAWA A., KODAMA Y., Solitons in Optical Communications, Clarendon Press, Oxford,1995.
[3] HASEGAWA A., Theory of information transfer in optical fibers: A tutorial review, Optical Fiber
Technology 10(2), 2004, pp. 150–170.
[4] MOUSSA SMADI, DERRADJI BAHLOUL, A compact split step Padé scheme for higher-order nonlinear
Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion, Computer
Physics Communications 182(2), 2011, pp. 366–371.
110
XIUJUN HE, KANG XIE
[5] WOO-PYO HONG, Modulation instability of optical waves in the high dispersive cubic–quintic
nonlinear Schrödinger equation, Optics Communications 213(1–3), 2002, pp. 173–182.
[6] TANEV S., PUSHKAROV D.I., Solitary wave propagation and bistability in the normal dispersion
region of highly nonlinear optical fibers and waveguides, Optics Communications 141(5–6), 1997,
pp. 322–328.
[7] SONESON J., PELEG A., Effect of quintic nonlinearity on soliton collisions in fibers, Physica D:
Nonlinear Phenomena 195(1–2), 2004, pp. 123–140.
[8] KONAR S., MANOJ MISHRA, JANA S., The effect of quintic nonlinearity on the propagation
characteristics of dispersion managed optical solitons, Chaos, Solitons and Fractals 29(4), 2006,
pp. 823–828.
[9] ILDAY F.Ö., WISE F.W., Nonlinearity management: A route to high-energy soliton fiber lasers,
Journal of the Optical Society of America B 19(3), 2002, pp. 470–476
Received April 3, 2011
in revised form July 19, 2011